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Chapter 4: Problem 28

Write the equation in logarithmic form. $$ 2^{5}=32 $$

Short Answer

Expert verified
log_2(32) = 5

Step by step solution

01

Identify the Elements

Recognize the base, exponent, and result from the given exponential equation. Here, the base is 2, the exponent is 5, and the result is 32.
02

Apply the Logarithmic Form

Remember that the logarithmic form of an exponential equation such as 2^5=32 is given by: log_base(result) = exponent. Therefore, we substitute base=2, result=32, and exponent=5.
03

Write the Logarithmic Equation

Using the formula from Step 2, we write log_2(32) = 5.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Logarithms
Logarithms might seem confusing at first, but they're just another way to express exponential equations. In essence, a logarithm asks the question: 'What power do we need to raise a given base to, in order to get a certain number?' For example, if we have the equation \(2^{5}=32\), we can rewrite it as a logarithmic equation: \(\text{log}_{2}(32)=5\). This means that if you raise the base 2 to the power of 5, you get 32.
Exponential Equations
Exponential equations are equations where the variable is in the exponent. For instance, in the equation \(2^{5}=32\), 2 is the base, 5 is the exponent, and 32 is the result. Exponential equations can always be converted into logarithmic form, which can often make solving them easier. To convert, remember the relationship: \(b^{e}=r\) converts to \(\text{log}_{b}(r)=e\). Understanding this conversion is key to working with both forms.
Base and Exponent
When working with exponential and logarithmic forms, it's crucial to identify the base and the exponent correctly. In the given problem \(2^{5}=32\):
  • The base is 2. This is the number that's being raised to a power.
  • The exponent is 5. This indicates how many times the base is multiplied by itself.
  • The result is 32. This is what you get after performing the exponentiation.
From this, the corresponding logarithmic form is \(\text{log}_{2}(32)=5\), showing that 2 raised to the power of 5 equals 32.

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Most popular questions from this chapter

According to the CIA's World Fact Book, in \(2010,\) the population of the United States was approximately 310 million with a \(0.97 \%\) annual growth rate. (Source: www.cia.gov) At this rate, the population \(P(t)\) (in millions) can be approximated by \(P(t)=310(1.0097)^{t}\), where \(t\) is the time in years since 2010 . a. Is the graph of \(P\) an increasing or decreasing exponential function? b. Evaluate \(P(0)\) and interpret its meaning in the context of this problem. c. Evaluate \(P(10)\) and interpret its meaning in the context of this problem. Round the population value to the nearest million. d. Evaluate \(P(20)\) and \(P(30)\). e. Evaluate \(P(200)\) and use this result to determine if it is reasonable to expect this model to continue indefinitely.

Explain why the \(f(x)=x^{2}\) is not an exponential function.

Suppose that \(\$ 50,000\) from a retirement account is invested in a large cap stock fund. After 20 yr, the value is \(\$ 194,809.67\). a. Use the model \(A=P e^{r t}\) to determine the average rate of return under continuous compounding. b. How long will it take the investment to reach onequarter million dollars? Round to 1 decimal place.

Use the model \(A=P\left(1+\frac{r}{n}\right)^{n t} .\) The variable A represents the future value of P dollars invested at an interest rate \(r\) compounded \(n\) times per year for \(t\) years. If 4000 is put aside in a money market account with interest reinvested monthly at 2.2%, find the time required for the account to earn 1000. Round to the nearest month.

The atmospheric pressure on an object decreases as altitude increases. If \(a\) is the height (in \(\mathrm{km}\) ) above sea level, then the pressure \(P(a)\) (in \(\mathrm{mmHg}\) ) is approximated by \(P(a)=760 e^{-0.13 a}\) a. Find the atmospheric pressure at sea level. b. Determine the atmospheric pressure at \(8.848 \mathrm{~km}\) (the altitude of \(\mathrm{Mt}\). Everest). Round to the nearest whole unit.
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